The singular value decomposition of the matrix \(A \in {m \times n}\) , factors it as \(A=U \Sigma V^{*}\) , where \(U\in {m \times m}\)  and \(V\in {n \times n}\)  are orthonormal matrices, and \(\Sigma \)  is an \(m \times n\)  real diagonal matrix with entries \(\sigma _{1}, \sigma _{2}, \cdots , \sigma _{n}\)  ordered such that \(\sigma _{1} \ge \sigma _{2} \ge \cdots \ge \sigma _{n} \ge 0\) . When \(A\)  is real, so are \(U\)  and \(V\) . The truncated SVD of \(A\)  is given by \(U_r \Sigma _r V^{\top }_r (r<\min \lbrace m,n\rbrace )\) , where the matrices \(U_{r}\)  and \(V_{r}\)  contain only the first \(r\)  columns of \(U\)  and \(V\) , and \(\Sigma _{r}\)  contains only the first \(r\)  singular values from \(\Sigma \) . According to the Eckart-Young theorem [1]}, the best rank \(r\)  approximation to \(A\)  in the spectral norm or Frobenius norm is given by
\(A_{[r]}=\sum _{i=1}^{r} \sigma _{i} u_{i} v_{i}^{\top }\) 
